How to use snap option in object mode?
Object Mode Snapping:


Center
When in Object mode, "Center" for snap means taking the current Pivot point setting of your selection, which also means the transformation center, not the bounding box center of your selection.
Median
As for "median", it means the same as the Pivot type Median.
Median means a sort of center of gravity of the selection. The behavior is most obvious with an  asymmetric selection:


There is one way to get the same result by using "Center" and "Median" snapping,  which is when using Center snapping mode, if your Pivot point setting is Median, then it behaves the same as when using "Median" snapping mode.
In any other case, they will not be same unless your selected objects are distributed perfectly even with each other.
Vert/edge/face


Vert/edge/face are only available in Edit mode. So, don't think about it in Object mode, when in object mode, unless you are using "Closet" as snap target (available for vertex). your snap target "Active" will only be the origin of the one you selected last. Furthermore, make sure you do have a object that is selected and activated, otherwise blender has nothing available to snap, but to use "Median" point as the alternative one. 

I hope I've tried my best to answer every point in your question. Of course, if any question, just throw it here, we'll try to offer further help. :)



P.S.: If you really need to snap certain element (vert/edge/face) in Object mode, you may consider switching to Edit Mode first, place/snap 3D cursor to wherever you want, then go back to Object Mode, using Center as snap target. Then switch Pivot point to 3D Cursor, which may behave as you expected.